On the structure of existence regions for sinks of the Hénon map

Zbigniew Galias, Warwick Tucker

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Abstract

An extensive search for stable periodic orbits (sinks) for the Hénon map in a small neighborhood of the classical parameter values is carried out. Several parameter values which generate a sink are found and verified by rigorous numerical computations. Each found parameter value is extended to a larger region of existence using a simplex continuation method. The structure of these regions of existence is investigated. This study shows that for the Hénon map, there exist sinks close to the classical case.

The goal of the theory of dynamical systems is to give a description of the long-term behaviour of a given system. Invariant sets such as fixed points, periodic points, Smale horseshoes, and more complicated, strange attractors are examples of sets that remain present after the transient behaviour of the system has settled down. Of these invariant sets, only the ones that are attracting are detectable in some physical sense. In this paper, we study the presence of stable periodic orbits (sinks) for the Hénon map. The main challenge is to locate such sinks as they have very small regions of existence. Here we demonstrate that there exist low-period sinks extremely close to the classical parameter values of the Hénon map. Our conclusion is that most numerical studies to this date do not display anything but transient behaviour and are therefore inconclusive as to the true nature of the long-term dynamics of the Hénon map.
Original languageEnglish
Article number013120
Number of pages14
JournalChaos: An Interdisciplinary Journal of Nonlinear Science
Volume24
Issue number1
DOIs
Publication statusPublished - 2014
Externally publishedYes

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